 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says a right triangle whose sides are 3 cm and 4 cm other than hypotenuse is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed choose value of pi as found appropriate. Now we know that volume of cone is equal to 1 by 3 pi r square h where r is the radius of the base of the cone and h is the height surface area of cone is r n where r is the radius and l is the slant height of the cone. So this is the key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Let in this figure a b c be the right angle triangle such that a b is equal to 3 cm and a c is equal to 4 cm and this is made to revolve about its hypotenuse. So a dash b is also equal to 3 cm and a dash c is 4 cm. So let a b c be the right angle triangle such that a b is equal to 3 cm and a c is equal to 4 cm. So by Pythagoras theorem we have b c square is equal to a b square plus a c square or b c square is equal to a b c cm so 3 square plus 4 square and this is equal to 9 plus 16 that is b c square is equal to 25 so b c is equal to 5 cm. Now let o b is equal to x cm and a o is equal to y cm. So this is x cm and this is y cm. So by Pythagoras theorem in triangle o a b we have k is equal to o b square plus o a square a b 3 cm so 3 square is equal to o b x cm x square plus o a is y cm y square therefore x square plus y square is equal to 9. Let us give this as number 1. Again in triangle o a c we have c square is equal to o c square plus a o square by Pythagoras theorem. Now a c is 4 cm so 4 square is equal to now b c is 5 cm and o b is x cm therefore o c is 5 minus x cm. So this is 5 minus x square plus y square because b c is 5 cm and o c is equal to b c minus o b that is 5 minus x cm. 5 minus x square plus y square is equal to 16. Let us give this as number 2. Now subtracting 1 from 2 we have 5 minus x square plus y square minus x square plus y square is equal to 16 minus 9. Now we will solve this equation so we have 5 minus x whole square plus y square minus x square minus y square is equal to 7 this implies 5 minus x whole square minus x square is equal to 7 this implies 5 minus x plus x into 5 minus x minus x is equal to 7 because a square minus b square is equal to a plus b into a minus b. So this implies 5 into 5 minus 2 x is equal to 7 and this implies 25 minus 10 x is equal to 7 this implies 10 x is equal to 25 minus 7 and this implies 10 x is equal to 18. Therefore x is equal to 18 upon 10 which is equal to 1.8. Now by putting the value of in equation 1 we have my equation 1 was x square plus y square is equal to 9 so 1.8 square plus y square is equal to 9 this implies now 1.8 square is 3.24 plus y square is equal to 9 this implies y square is equal to 9 minus 3.24 and this implies y square is equal to 5.76 therefore y is equal to under root of 5.76 which is equal to 2.4. Therefore we have is equal to 2.4 centimeter o b is equal to 1.8 centimeter and o c which is equal to b c minus o b is equal to 5 minus 1.8 and this is equal to 3.2 centimeter. Now when the right triangle is made to revolve about its hypotenuse double cone is formed. We have to find the volume and the surface area of the double cone. Now volume of the double cone is equal to volume of cone a a dash, volume of cone b a a dash and this is equal to now volume of cone c a a dash is 1 by 3 pi r square. Now r is a o into h h is o c plus volume of cone b a a dash that is 1 by 3 into pi into r square h is o b. So this is equal to 1 by 3 into pi into o a is 2.4 centimeter so 2.4 square into 3.2 plus 1 by 3 pi into 2.4 square into 1.8 centimeter cube and this is equal to let us take 1 by 3 pi into 2.4 square common we have this is 3.2 plus 1.8 centimeter cube and this is equal to 1 by 3 take pi is 22 upon 7 into 2.4 into 2.4 into 5 centimeter cube and this is equal to 633.6 upon 21 centimeter cube and this is again equal to 30.17 centimeter cube. So this is the volume of the double cone that is volume of double cone is equal to 30.17 centimeter cube. Now surface area of double cone is equal to curve surface area of cone a a dash plus curve surface area of cone b a a dash and this is equal to now curve surface area of cone c a a dash is pi into r which is o a into l which is ac plus curve surface area of cone b a a dash is pi into r o a into l a b and this is equal to pi into o a is 2.4 centimeter and ac is 4 centimeter plus pi into o a is 2.4 centimeter into a b and this is equal to let us take pi into 2.4 common we have 4 plus 3 inside the bracket let us put pi is equal to 22 upon 7 into 2.4 into 7 centimeter square and this is equal to 52.8 centimeter square therefore surface area of double cone is equal to 52.8 centimeter square. Hence the answer for the above question is volume of the double cone is 30.17 centimeter cube and surface area of double cone is 52.8 centimeter square I hope the solution is clear to you buy and take care